AP Precalculus · Summary

Unit 1 summary

1. Average rate of change is $\frac{f(b)-f(a)}{b-a}$. In a table with equal steps, growing rates suggest concave up and shrinking rates suggest concave down.
2. A degree $n$ polynomial has at most $n$ real zeros and at most $n-1$ local extrema. When the coefficients are real, nonreal zeros come in conjugate pairs.
3. Odd multiplicity crosses the $x$-axis; even multiplicity touches and turns back.
4. A continuous function that changes sign between two inputs has a zero between them (Intermediate Value Theorem).
5. The leading term sets end behavior. For a rational function, compare degrees to find the horizontal or slant asymptote.
6. Factor first: a factor that cancels leaves a hole, and one left in the denominator gives a vertical asymptote.
7. In $a\,f(b(x-h))+k$, changes outside act on outputs as written; changes inside act on inputs in reverse.